Question

Space between the plates of a parallel plate capacitor of plate area 4 cm$^2$ and separation of $ d = 1.77 \, \text{mm} $, is filled with uniform dielectric materials with dielectric constants (3 and 5) as shown in figure. Another capacitor of capacitance 7.5 pF is connected in parallel with it. The effective capacitance of this combination is ____ pF.

Hint:

When two dielectric materials are used in parallel plate capacitors, calculate the individual capacitances for each dielectric, and then treat them as separate capacitors in series or parallel as per the configuration.

Answer 1

Correct Answer: 15

Given: - Plate area of the capacitor, \( A = 4 \, \text{cm}^2 = 4 \times 10^{-4} \, \text{m}^2 \),
- Separation between the plates, \( d = 1.77 \, \text{mm} = 1.77 \times 10^{-3} \, \text{m} \),
- Dielectric constants for the two regions, \( k_1 = 5 \) and \( k_2 = 3 \), - Capacitance of an additional parallel capacitor, \( C_3 = 7.5 \, \text{pF} \),
- Permittivity of free space, \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \).

Step 1: Calculate the capacitance for each section with the dielectric constants. For the first section (with dielectric constant \( k_1 = 5 \)): The capacitance is given by: \[ C_1 = \frac{\varepsilon_0 A}{d/2} \cdot k_1 \] Substitute the values: \[ C_1 = \frac{(8.85 \times 10^{-12}) \times (4 \times 10^{-4})}{1.77 \times 10^{-3}/2} \cdot 5 \] First calculate the denominator \( d/2 = \frac{1.77 \times 10^{-3}}{2} = 8.85 \times 10^{-4} \, \text{m} \). Now substitute: \[ C_1 = \frac{8.85 \times 10^{-12} \times 4 \times 10^{-4}}{8.85 \times 10^{-4}} \cdot 5 = 4 \times 10^{-12} \times 5 = 20 \times 10^{-12} \, \text{F} = 20 \, \text{pF} \] For the second section (with dielectric constant \( k_2 = 3 \)): Using the same formula: \[ C_2 = \frac{\varepsilon_0 A}{d/2} \cdot k_2 \] Substitute the values: \[ C_2 = \frac{(8.85 \times 10^{-12}) \times (4 \times 10^{-4})}{1.77 \times 10^{-3}/2} \cdot 3 \] \[ C_2 = \frac{8.85 \times 10^{-12} \times 4 \times 10^{-4}}{8.85 \times 10^{-4}} \cdot 3 = 4 \times 10^{-12} \times 3 = 12 \times 10^{-12} \, \text{F} = 12 \, \text{pF} \]
Step 2: Find the total capacitance of the two dielectric regions in series. The two dielectric regions are in series, so the total capacitance \( C_{\text{total}} \) is given by: \[ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} \] Substitute the values for \( C_1 \) and \( C_2 \): \[ \frac{1}{C_{\text{total}}} = \frac{1}{20 \, \text{pF}} + \frac{1}{12 \, \text{pF}} \] \[ \frac{1}{C_{\text{total}}} = \frac{3}{60 \, \text{pF}} + \frac{5}{60 \, \text{pF}} = \frac{8}{60 \, \text{pF}} = \frac{2}{15 \, \text{pF}} \] Thus, \[ C_{\text{total}} = \frac{15}{2} \, \text{pF} = 7.5 \, \text{pF} \]
Step 3: Add the capacitance of the additional parallel capacitor. The total capacitance of the system is the sum of the series capacitance and the parallel capacitance \( C_3 \): \[ C_{\text{effective}} = C_{\text{total}} + C_3 \] \[ C_{\text{effective}} = 7.5 \, \text{pF} + 7.5 \, \text{pF} = 15 \, \text{pF} \] Thus, the effective capacitance is \( \boxed{15 \, \text{pF}} \).